Basic Math Examples

$t (2 t - 1) + 3 = 3 (3 t + 2)$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$t (2 t) + t \cdot - 1 + 3 = 3 (3 t + 2)$

Step 1.2

Rewrite using the commutative property of multiplication.

$2 t \cdot t + t \cdot - 1 + 3 = 3 (3 t + 2)$

Step 1.3

Move $- 1$ to the left of $t$ .

$2 t \cdot t - 1 \cdot t + 3 = 3 (3 t + 2)$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $t$ by $t$ by adding the exponents.

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Step 1.4.1.1

Move $t$ .

$2 (t \cdot t) - 1 \cdot t + 3 = 3 (3 t + 2)$

Step 1.4.1.2

Multiply $t$ by $t$ .

$2 t^{2} - 1 \cdot t + 3 = 3 (3 t + 2)$

Step 1.4.2

Rewrite $- 1 t$ as $- t$ .

$2 t^{2} - t + 3 = 3 (3 t + 2)$

Step 2

Simplify $3 (3 t + 2)$ .

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Step 2.1

Apply the distributive property.

$2 t^{2} - t + 3 = 3 (3 t) + 3 \cdot 2$

Step 2.2

Multiply.

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Step 2.2.1

Multiply $3$ by $3$ .

$2 t^{2} - t + 3 = 9 t + 3 \cdot 2$

Step 2.2.2

Multiply $3$ by $2$ .

$2 t^{2} - t + 3 = 9 t + 6$

Step 3

Move all terms containing $t$ to the left side of the equation.

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Step 3.1

Subtract $9 t$ from both sides of the equation.

$2 t^{2} - t + 3 - 9 t = 6$

Step 3.2

Subtract $9 t$ from $- t$ .

$2 t^{2} - 10 t + 3 = 6$

Step 4

Move all terms to the left side of the equation and simplify.

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Step 4.1

Subtract $6$ from both sides of the equation.

$2 t^{2} - 10 t + 3 - 6 = 0$

Step 4.2

Subtract $6$ from $3$ .

$2 t^{2} - 10 t - 3 = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = 2$ , $b = - 10$ , and $c = - 3$ into the quadratic formula and solve for $t$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (2 \cdot - 3)}}{2 \cdot 2}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 10$ to the power of $2$ .

$t = \frac{10 \pm \sqrt{100 - 4 \cdot 2 \cdot - 3}}{2 \cdot 2}$

Step 7.1.2

Multiply $- 4 \cdot 2 \cdot - 3$ .

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$t = \frac{10 \pm \sqrt{100 - 8 \cdot - 3}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $- 3$ .

$t = \frac{10 \pm \sqrt{100 + 24}}{2 \cdot 2}$

Step 7.1.3

Add $100$ and $24$ .

$t = \frac{10 \pm \sqrt{124}}{2 \cdot 2}$

Step 7.1.4

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 7.1.4.1

Factor $4$ out of $124$ .

$t = \frac{10 \pm \sqrt{4 (31)}}{2 \cdot 2}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$t = \frac{10 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 2}$

Step 7.1.5

Pull terms out from under the radical.

$t = \frac{10 \pm 2 \sqrt{31}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$t = \frac{10 \pm 2 \sqrt{31}}{4}$

Step 7.3

Simplify $\frac{10 \pm 2 \sqrt{31}}{4}$ .

$t = \frac{5 \pm \sqrt{31}}{2}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{5 + \sqrt{31}}{2}, \frac{5 - \sqrt{31}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$t = \frac{5 + \sqrt{31}}{2}, \frac{5 - \sqrt{31}}{2}$

Decimal Form:

$t = 5.28388218 \dots, - 0.28388218 \dots$